. Therefore the direction vector of the first line is with
. Therefore the direction vector of the second line is with
2. The point of intersection is . Therefore the angle bisector has to pass through this point too.
3. The direction of the angle bisector is the sum of the unit vectors of the two direction vectors:
The second angle bisector is perpendicular to the first one thus it has the direction
4. You can simplify those direction vectors to and
5. Now use the point direction form of the equation of a line to determine the equations of the two angle bisectors.
EDIT: I've attached a sketch of the two lines and the two angle bisectors